Quark matter in neutron stars
Mark AlfordWashington University in St. Louis
Outline
I Quarks at high densityConfined, quark-gluon plasma, color superconducting
II Color superconductivityColor-flavor locking (CFL), and beyond
III Compact starsSignatures of the presence of quark matter
IV Looking to the future
M. Alford, K. Rajagopal, T. Schafer, A. Schmitt, arXiv:0709.4635 (RMP)A. Schmitt, arXiv:1001.3294 (Springer Lecture Notes)
I. Quarks at high density
Quarks: Building blocks of matter
& electronsatom: nucleus
3 quarks boundby color force
neutron/proton:
,Quantum (QCD)Chromo Dynamics
Quarks have color andflavor (“up” or “down”)
proton: uud, uud, uudneutron: udd, udd, udd
Phase Transitions
When you heat up or compress matter, theatoms reconfigure themselves:Phase transitions between solid, liquid, andgas. Ice
p
TWatervapor
1atm
Liquidwater
At super-high temperatures or densities, when the nuclei are constantlybashed around or remorselessly crushed together, do quarks reconfigurethemselves?
T ∼ 150 MeV ∼ 1012 K
ρ ∼ 300 MeV/fm3 ∼ 1017 kg/m3At such a density, a oil super-tanker is 1mm3 in size.
Where might this occur?• supernovas, neutron stars;
• Brookhaven (AGS, RHIC); CERN (SPS, LHC)
Conjectured QCD phase diagram
superconductingquark matter
= color−
liq
T
µ
gas
QGP
CFL
nuclear
/supercondsuperfluid
compact star
non−CFL
heavy ioncollider
hadronic
heavy ion collisions: chiral critical point and first-order linecompact stars: color superconducting quark matter core
Signatures of quark matter in compact stars
Observable ← Microphysical properties(and neutron star structure)
← Phases of dense matter
mass, radius eqn of state
spindown(spin freq, age)
bulk viscosityshear viscosity
cooling(temp, age)
heat capacityneutrino emissivitythermal cond.
glitches(superfluid,crystal)
shear modulusvortex pinning
energy
Color superconductivityAt sufficiently high density and low tempera-ture, there is a Fermi sea of almost free quarks.
nuclear
non−CFL
CFLliq
T
µ
gas
QGP
µ = EF
E
p
F = E − µN
dF
dN= 0
But quarks have attractiveQCD interactions.
Any attractive quark-quark interaction causes pairing instability of the
Fermi surface: BCS mechanism of superconductivity.BCS in quark matter: Ivanenko and Kurdgelaidze, Lett. Nuovo Cim. IIS1 13 (1969).
What is a condensate of Cooper pairs?
p
E
µ
|φ0〉 =∏
p
(cos(θAp) + sin(θAp) a†(p)a†(−p)
)(
cos(θBp) + sin(θBp) b†(p)b†(−p))× |Fermi sea〉
|φ0〉, not |Fermi sea〉, is the ground state.
Physical consequences of Cooper pairing
Changes low energy excitations, affecting transport properties.
I Spontaneous breaking of global symmetries ⇒ Goldstone bosons,massless degrees of freedom that dominate low energy behavior.E.g.: Superfluidity
I Spontaneous breaking of local (gauged) symmetries: massivegauge bosons, exclusion of magnetic fields (Meissner effect).E.g.: Superconductivity
I Gap in fermion spectrum.
Adding a fermion near the Fermi surfacenow costs energy because it disrupts thecondensate.
a†p(cos θ + sin θ a†pa†−p) = cos θ a†p
E
p
∆
particlehole
qua
si holequa
rtpaicle
si
Interactions between Quarks
Dominant interaction between quarks is the strong interaction,mediated by exchange of gluons that couple to “color” charge (QCD).
Properties of QCD
I Short distances, r 1 fm, asymptotically free :gauge coupling g 1, single gluon exchange dom-inates, the theory is analytically tractable.
g
g
I Long distances r > 1 fm, QCD confines : colorelectric fields form flux tubes, only color-neutralstates, baryons and mesons, exist. baryon meson
I At low temperature (T . 170 MeV),
Chiral (left-right) symmetry is broken : colorforce can’t turn a LH quark to RH, but ourvacuum is full of qLqR pairs
LR
L R
LR
Interactions between Quarks
Dominant interaction between quarks is the strong interaction,mediated by exchange of gluons that couple to “color” charge (QCD).
Properties of QCD
I Short distances, r 1 fm, asymptotically free :gauge coupling g 1, single gluon exchange dom-inates, the theory is analytically tractable.
g
g
I Long distances r > 1 fm, QCD confines : colorelectric fields form flux tubes, only color-neutralstates, baryons and mesons, exist. baryon meson
I At low temperature (T . 170 MeV),
Chiral (left-right) symmetry is broken : colorforce can’t turn a LH quark to RH, but ourvacuum is full of qLqR pairs
LR
L R
LR
Handling QCD at high density
Lattice: “Sign problem”—negative probabilities
SUSY: Statistics crucial to quark Fermi surface
large N: Quarkyonic phase?
pert: Applicable far beyond nuclear density.Neglects confinement and instantons.
NJL: Model, applicable at low density.Follows from instanton liquid model.
EFT: Effective field theory for lightest degrees of freedom.“Parameterization of our ignorance”: assume a phase, guesscoefficients of interaction terms (or match to pert theory),obtain phenomenology.
Color superconducting phases
Attractive QCD interaction ⇒ Cooper pairing of quarks.
We expect pairing between different flavors .
Quark Cooper pair: 〈qαiaqβ
jb〉color α, β = r , g , bflavor i , j = u, d , sspin a, b =↑, ↓
Each possible BCS pairing pattern P is an 18× 18 color-flavor-spinmatrix
〈qαiaqβ
jb〉1PI = ∆P Pαβij ab
The attractive channel is:
space symmetric [s-wave pairing]color antisymmetric [most attractive]spin antisymmetric [isotropic]
⇒ flavor antisymmetric
Initially we will assume the most symmetric case, where all three flavorsare massless.
High-density QCD calculations• Guess a color-flavor-spin pairing pattern P• to obtain gap ∆P , minimize free energy Ω with respect to ∆P
(imposing color and electric neutrality)
∂Ω
∂∆P= 0
∂Ω
∂µi= 0
The pattern with the lowest Ω(∆P) wins!
1. Weak-coupling methods. First-principles calculations direct fromQCD Lagrangian, valid in the asymptotic regime, currentlyµ & 106 MeV.
2. Nambu–Jona-Lasinio models, ie quarks with four-fermion couplingbased on instanton vertex, single gluon exchange, etc. This is asemi-quantitative guide to physics in the compact star regimeµ ∼ 400 MeV, not a systematic approximation to QCD.
NJL gives ∆ ∼ 10−100 MeV at µ ∼ 400 MeV.
Gap equation in a simple NJL model
Minimize free energy wrt ∆:
1 =8K
π2
∫ Λ
0
p2dp
1√
∆2 + (p − µ)2
RHS
∆
1
Note BCS divergence as ∆→ 0: there is always a solution, for anyinteraction strength K and chemical potential µ.Roughly,
1 ∼ Kµ2 ln (Λ/∆)
⇒ ∆ ∼ Λ exp
(− 1
Kµ2
)Superconducting gap is non-perturbative.
Color supercond. in 3 flavor quark matter
Color-flavor locking (CFL)Equal number of colors and flavors gives a special pairing pattern(Alford, Rajagopal, Wilczek, hep-ph/9804403)
〈qαi qβ
j 〉 ∼ δαi δβj − δαj δβi = εαβnεijn
color α, βflavor i , j
This is invariant under equal and oppositerotations of color and (vector) flavor
SU(3)color × SU(3)L × SU(3)R︸ ︷︷ ︸⊃ U(1)Q
×U(1)B → SU(3)C+L+R︸ ︷︷ ︸⊃ U(1)Q
×Z2
I Breaks chiral symmetry, but not by a 〈qq〉 condensate.I There need be no phase transition between the low and high
density phases: (“quark-hadron continuity”)I Unbroken “rotated” electromagnetism, Q, photon-gluon mixture.
Color-flavor-locked (“CFL”) quark pairing
Q 0 0 0 −1 +1 −1 +1 0 0u d s d u s u s d
u ∆ ∆d ∆ ∆s ∆ ∆d −∆u −∆s −∆u −∆s −∆d −∆
Conjectured QCD phase diagram
liq
T
µ
gas
QGP
CFL
nuclear
/supercondsuperfluid
compact star
non−CFL
heavy ioncollider
hadronic
III. Quark matter in compact stars
Where in the universe is color-superconducting quark matter most likelyto exist? In compact stars.
A quick history of a compact star.
A star of mass M & 10M burns Hydrogen by fusion, ending up withan Iron core. Core grows to Chandrasekhar mass, collapses ⇒supernova. Remnant is a compact star:
mass radius density initial temp∼ 1.4M O(10 km) & ρnuclear ∼ 30 MeV
The star cools by neutrino emission for the first million years.
The real world: Ms and neutrality
In the real world there are three factors that combine to oppose pairingbetween different flavors.
1. Strange quark mass is not infinite nor zero, but intermediate. Itdepends on density, and ranges between about 500 MeV in thevacuum and about 100 MeV at high density.
2. Neutrality requirement. Bulk quark matter must be neutralwith respect to all gauge charges: color and electromagnetism.
3. Weak interaction equilibration. In a compact star there is timefor weak interactions to proceed: neutrinos escape and flavor is notconserved.
These factors favor different Fermi momenta for different flavors whichobstructs pairing between different flavors.
Mismatched Fermi surfaces oppose Cooperpairing
Fpd
Fup
u and d quarks near their Fermisurfaces cannot have equal andopposite momenta.
〈u(k)d(−k)〉 condensate isenergetically penalized.
The strange quark mass is the causeof the mismatch:
pFd − pFu ≈ pFu − pFs ≈M2
s
4µ
Cooper pairing vs. the strange quark mass
Unpaired
blue
s
p
red green
F
u
dMs
2
4µ
2SC pairing
blue
s
p
red green
F
u
d
CFL pairing
blue
p
red green
F
CFL: Color-flavor-locked phase, favored at the highest densities.
〈qαi qβ
j 〉 ∼ δαi δβj − δαj δβi = εαβNεijN
2SC: Two-flavor pairing phase. May occur at intermediate densities.
〈qαi qβ
j 〉 ∼ εαβ3εij3 ∼ (rg − gr)(ud − du)
or: CFL with kaon condensation (CFL-K 0),crystalline phase (LOFF), p-wave “meson” condensates,single-flavor pairing (color-spin locking, ∼liq 3He-B).
Phases of quark matter, again
liq
T
µ
gas
QGP
CFL
nuclear
/supercondsuperfluid
compact star
non−CFL
heavy ioncollider
hadronic
NJL model, uniform phases only
µB/3 (MeV)T(M
eV)
550500450400350
60
50
40
30
20
10
0
g2SC
NQ
NQ
2SC
uSCguSC →
CFL
← gCFL
CFL-K0
p2SC−→χSB
t
t tt
Warringa, hep-ph/0606063
But there are also non-uniform phases, such as the crystalline(“LOFF”/”FFLO”) phase. (Alford, Bowers, Rajagopal, hep-ph/0008208)
Crystalline (LOFF) superconductivityWhen the Fermi momenta are such that one flavor of quark is just barely
excluded from pairing with another, it may be favorable to make pairs with a
net momentum, so each flavor can be close to its Fermi surface.
q
p
p
Every quark pair in the condensate has the same nonzero total momentum
2q (single plane wave LOFF).
Free energy comparison of phases
Assuming ∆CFL = 25 MeV.
0 50 100 150 200M
2
S/µ [MeV]
-50
-40
-30
-20
-10
0
En
erg
y D
iffe
ren
ce
[10
6 M
eV4]
gCFL
CFL
unpaired
2SC
g2SC
2PW
CubeX2Cube45z
CFL-K0
curCFL
CFL-K 0 K 0 condensatecurCFL K 0 cond current2PW LOFF, 2-plane-waveCubeX LOFF crystal, G-L approx2Cube45z LOFF crystal, G-L approx
(Alford, Rajagopal, Schafer, Schmitt, arXiv:0709.4635)
Curves for CubeX and 2Cube45z use G-L approx far from its area ofvalidity: favored phase at M2
s ∼ 4µ∆ remains uncertain.
Signatures of quark matter in compact stars
Observable ← Microphysical properties(and neutron star structure)
← Phases of dense matter
Property Nuclear phase Quark phase
mass, radius eqn of stateknownup to nsat
unknown, can beparameterized
spindown(spin freq, age)
bulk viscosityshear viscosity
Depends onphase:
n p en p e, µn p e,Λ, Σ−
n superfluidp supercondπ condensateK condensate
Depends onphase:
unpairedCFLCFL-K 0
2SCCSLLOFF1SC. . .
cooling(temp, age)
heat capacityneutrino emissivitythermal cond.
glitches(superfluid,crystal)
shear modulusvortex pinning
energy
Signatures of quark matter in compact stars
Observable ← Microphysical properties(and neutron star structure)
← Phases of dense matter
Property Nuclear phase Quark phase
mass, radius eqn of stateknownup to nsat
unknown, can beparameterized
spindown(spin freq, age)
bulk viscosityshear viscosity
Depends onphase:
n p en p e, µn p e,Λ, Σ−
n superfluidp supercondπ condensateK condensate
Depends onphase:
unpairedCFLCFL-K 0
2SCCSLLOFF1SC. . .
cooling(temp, age)
heat capacityneutrino emissivitythermal cond.
glitches(superfluid,crystal)
shear modulusvortex pinning
energy
Discovery of a 2M mass neutron star
7 8 9 10 11 12 13 14 15Radius (km)
0.0
0.5
1.0
1.5
2.0
2.5
Mass
(so
lar)
AP4
MS0
MS2
MS1
MPA1
ENG
AP3
GM3PAL6
GS1
PAL1
SQM1
SQM3
FSU
GR
P <
causality
rotation
J1614-2230
J1903+0327
J1909-3744
Double NS Systems
Nucleons Nucleons+ExoticStrange Quark Matter
M = 1.97± 0.04 M
Demorest et al,
Nature 467,
1081 (2010).
Can quark matter be the favored phase at high density?
Constraints on the quark matter EoSGeneric ansatz: ε(p) = εcrit+∆ε + c−2
QM(p − pcrit)
QM + Soft Nuclear Matter
2.3M ๏
2.1M
๏
2.0M๏
HLPS+QM EoSncrit = 2.0n0
ncrit = 4.0n0
2.2M๏
2.0M๏
c QM
2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Δε/εcrit
0 0.2 0.4 0.6 0.8 1
QM + Hard Nuclear Matter
2.4M๏ 2.2M๏
2.0M
๏
NL3+QM EoSncrit = 1.5n0
ncrit = 2.0n0
2.4M
๏ 2.2M๏
2.0M๏
c QM
2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Δε/εcrit
0 0.25 0.5 0.75 1 1.25 1.5
Alford, Han, Prakash, arXiv:1302.4732
1. Observations can constrain QM EoS but not rule out generic QM
2. Constraints depend on NM EoS up to transition density
r-modes: gravitational spin-down ofcompact stars
An r-mode is a quadrupole flow that emitsgravitational radiation. It becomes unsta-ble (i.e. arises spontaneously) when a starspins fast enough, and if the shear and bulkviscosity are low enough.
Side viewPolar view
mode pattern
star
Andersson gr-qc/9706075
Friedman and Morsink gr-qc/9706073
Constraints from r-modes: old stars (1)
¬
¬
¬¬
¬
¬
¬
¬
¬
¬
¬
¬ ¬
105 106 107 108 109 1010 10110
200
400
600
800
1000
T @KD
f@H
zD
Regions above curves are“forbidden” becauseviscosity is too low to holdback the r -modes.Only viscous damping included.
Data for accreting pulsarsin binary systems(LMXBs) vs instabilitycurves for nuclear andhybrid stars.
(Schwenzer, arXiv:1212.5242;Haskell, Degenaar, Ho,arXiv:1201.2101)
Need more than nuclear matter viscous damping
Constraints from r-modes: old stars (2)
¬
¬
¬¬
¬
¬
¬
¬
¬
¬
¬
¬ ¬
105 106 107 108 109 10100
200
400
600
800
T @KD
f@H
zD
Nuclear, viscous dampingonly
Nuclear with somecore-crust friction
Nuclear with maximumcore-crust friction
Free quarks
Quarks with non-Fermicorrections
(Schwenzer, unpublished)
Need something beyond the simple nuclear matter model
IV. Looking to the future
I Neutron-star phenomenology of color superconducting quarkmatter:
I Are there any other r-mode damping mechanisms?I neutrino emissivity and coolingI structure: nuclear-quark interface (gravitational waves?)I color supercond. crystalline phase (glitches) (gravitational waves?)I CFL: vortices but no flux tubes; stability of vortices. . .
I More general questions:I instability of gapless phases; better treatment of LOFFI role of large magnetic fieldsI better weak-coupling calculationsI better models of quark matter: Functional RG, Schwinger-DysonI solve the sign problem and do lattice QCD at high density.